Rips
class¶Initializes Rips
with the given distances whose main purpose
is to return the distance of two points given their indices. See
Distances below.
Calls functor with every simplex in the k-skeleton of the Rips complex (max). If seq is provided, then the complex is restricted to the vertex indices in the sequence.
Calls functor with every coface of the vertex v in the k-skeleton of the Rips complex (max). If seq is provided, then the complex is restricted to the vertex indices in the sequence.
Calls functor with every coface of the edge (u, v) in the k-skeleton of the Rips complex (max). If seq is provided, then the complex is restricted to the vertex indices in the sequence.
Compares simplices s1 and s2 with respect to their ordering in the Rips complex. Note that like Python’s built in cmp this is a three possible outsome comparison (-1,0,1) for (, respectively).
Returns the size of simplex s, i.e. the length of its longest edge.
An instance of distances passed to the constructor of Rips
should
know its length and the distances between the points. The length should be
retrievable via len(distance)
and it determines how many points the complex
is built on. The distances between the points are inferred by the class
Rips
by calling distances with a pair of vertices as arguments.
For example, the following class represents 10 points on an integer lattice:
class Distances:
def __len__(self):
return 10
def __call__(self, x, y):
return math.fabs(y-x)
The bindings expose a C++ class as a Python class PairwiseDistances
to deal with
explicit points in a Euclidean space. In pure Python it could be defined as
follows (in fact it used to be a pure Python class, and one may still find it in
bindings/python/dionysus/distances.py
; its performance is much slower
than its pure C++ analog):
class PairwiseDistances:
def __init__(self, points, norm = l2):
self.points = points
self.norm = norm
def __len__(self):
return len(self.points)
def __call__(self, p1, p2):
return self.norm([x - y for (x,y) in zip(self.points[p1], self.points[p2])])
Another distances class is available that speeds up the computation of the Rips
complex at the expense of the memory usage: ExplicitDistances
. It is
initialized with an instance of any class that behaves like a distances class,
and it stores all of its distances explicitly to not have to recompute them in
the future:
distances = PairwiseDistances(points)
distances = ExplicitDistances(distances)
With PairwiseDistances
being a C++ class, and
ExplicitDistances
being pure Python, the speed-up seems minor.
The following example reads in points from a file, and fills the list
simplices with the simplices of the 2-skeleton of the Rips complex built on
those vertices with distance cutoff parameter 50. Subsequently it computes the
persistence of the resulting filtration (defined by rips.cmp
):
points = [for p in points_file('...')]
distances = PairwiseDistances(points)
rips = Rips(distances)
simplices = Filtration()
rips.generate(2, 50, simplices.append)
simplices.sort(rips.cmp)
p = StaticPersistence(simplices)
p.pair_simplices()
Essentially the same example is implemented in
examples/rips/rips-pairwise.py
, although it reads the k and max
parameters for the Rips complex on the command line, and uses a trick to speed
up the computation.